Baking to Filmic or Filmic Log in Blender 2.9?
When you save the baked image check Save as Render to apply the render view transformation.
Update
The PNG standard is an 8/16-bit integer format.  To preserve HDR information you must use a file format that supports floating point, which means Radiance HDR or OpenEXR in Blender's case.
https://docs.blender.org/manual/en/latest/files/media/image_formats.html